Everything is illuminated

TL;DR

This paper investigates geometrical properties of translation surfaces, such as blocking and illumination, using dynamics of SL(2,R) actions, and extends key results to settle longstanding conjectures in the field.

Contribution

It characterizes surfaces with finite and bounded blocking properties and extends illumination results to non-lattice surfaces, settling a major conjecture.

Findings

Characterized surfaces with finite blocking property.

Characterized surfaces with bounded blocking property.

Extended illumination results to all translation surfaces, removing previous restrictions.

Abstract

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the SL(2,R)-action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert-Schmoll-Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture. Our results crucially rely on the recent breakthrough results of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, and on related results of Wright.